Solving General Joint Block Diagonalization Problem via Linearly Independent Eigenvectors of a Matrix Polynomial

Abstract

In this paper, we consider the exact/approximate general joint block diagonalization (GJBD) problem of a matrix set $\{A_i\}_{i=0}^p$ ($p\ge 1$), where a nonsingular matrix $W$ (often referred to as diagonalizer) needs to be found such that the matrices $W^{H}A_iW$'s are all exactly/approximately block diagonal matrices with as many diagonal blocks as possible. We show that the diagonalizer of the exact GJBD problem can be given by $W=[x_1, x_2, \dots, x_n]Π$, where $Π$ is a permutation matrix, $x_i$'s are eigenvectors of the matrix polynomial $P(λ)=\sum_{i=0}^pλ^i A_i$, satisfying that $[x_1, x_2, \dots, x_n]$ is nonsingular, and the geometric multiplicity of each $λ_i$ corresponding with $x_i$ equals one. And the equivalence of all solutions to the exact GJBD problem is established. Moreover, theoretical proof is given to show why the approximate GJBD problem can be solved similarly to the exact GJBD problem. Based on the theoretical results, a three-stage method is proposed and numerical results show the merits of the method.